Stochastic logarithm explained

In stochastic calculus, stochastic logarithm of a semimartingale

Y such that Y ≠ 0 and Y _{- ≠ 0} is the semimartingale X given by^[1] $$dX\_t=\backslash frac,\backslash quad\; X\_0=0.$$In layperson's terms, stochastic logarithm of Y measures the cumulative percentage change in Y .

Notation and terminology

The process

X obtained above is commonly denoted l{L}(Y) . The terminology stochastic logarithm arises from the similarity of l{L}(Y) to the natural logarithm log (Y) : If Y is absolutely continuous with respect to time and Y ≠ 0 , then X solves, path-by-path, the differential equation $$\backslash frac\; =\; \backslash frac,$$whose solution is X =log |Y| -log |Y _{0|} .

General formula and special cases

• Without any assumptions on the semimartingale

Y (other than Y ≠ 0 , Y _{- ≠} 0 ), one has$$\backslash mathcal(Y)\_t\; =\; \backslash log\backslash Biggl|\backslash frac\backslash Biggl|+\backslash frac12\backslash int\_0^t\backslash frac+\backslash sum\_\backslash Biggl(\backslash log\backslash Biggl|\; 1\; +\; \backslash frac\; \backslash Biggr|-\backslash frac\backslash Biggr),\backslash qquad\; t\backslash ge0,$$where [Y] ^{c} is the continuous part of quadratic variation of Y and the sum extends over the (countably many) jumps of Y up to time t .

• If

Y is continuous, then $$\backslash mathcal(Y)\_t\; =\; \backslash log\backslash Biggl|\backslash frac\backslash Biggl|+\backslash frac12\backslash int\_0^t\backslash frac,\backslash qquad\; t\backslash ge0.$$In particular, if Y is a geometric Brownian motion, then X is a Brownian motion with a constant drift rate.

• If

Y is continuous and of finite variation, then$$\backslash mathcal(Y)\; =\; \backslash log\backslash Biggl|\backslash frac\backslash Biggl|.$$Here Y need not be differentiable with respect to time; for example, Y can equal 1 plus the Cantor function.

Properties

• Stochastic logarithm is an inverse operation to stochastic exponential: If

\Delta X ≠ -1 , then l{L}( l{E}(X)) = X - X _{0} . Conversely, if Y ≠ 0 and Y _{- ≠} 0 , then l{E}( l{L}(Y)) = Y/Y _{0} .

• Unlike the natural logarithm

log (Y _{t)} , which depends only of the value of Y at time t , the stochastic logarithm l{L}(Y) _{t} depends not only on Y _{t} but on the whole history of Y in the time interval [ 0 ,t] . For this reason one must write l{L}(Y) _{t} and not l{L}(Y _{t)} .

• Stochastic logarithm of a local martingale that does not vanish together with its left limit is again a local martingale.
• All the formulae and properties above apply also to stochastic logarithm of a complex-valued

Y .

• Stochastic logarithm can be defined also for processes

Y that are absorbed in zero after jumping to zero. Such definition is meaningful up to the first time that Y reaches 0 continuously.^[2]

Useful identities

• Converse of the Yor formula: If

Y ^{(1)} ,Y ^{(2)} do not vanish together with their left limits, then$$\backslash mathcal\backslash bigl(Y^Y^\backslash bigr)\; =\; \backslash mathcal\backslash bigl(Y^\backslash bigr)\; +\; \backslash mathcal\backslash bigl(Y^\backslash bigr)\; +\; \backslash bigl[\backslash mathcal\{L\}\backslash bigl(Y^\{(1)\}\backslash bigr),\backslash mathcal\{L\}\backslash bigl(Y^\{(2)\}\backslash bigr)\backslash bigr].$$

• Stochastic logarithm of

1 / l{E}(X) :^[3] If \Delta X ≠ -1 , then$$\backslash mathcal\backslash biggl(\backslash frac\backslash biggr)\_t\; =\; X\_0-X\_t-[X]^c\_t+\backslash sum\_\backslash frac.$$

Applications

• Girsanov's theorem can be paraphrased as follows: Let

Q be a probability measure equivalent to another probability measure P . Denote by Z the uniformly integrable martingale closed by Z _{infty} = dQ/dP . For a semimartingale U the following are equivalent:

• 1. Process

U is special under Q .

• 1. Process

U + [U, l{L}(Z)] is special under P .

• + If either of these conditions holds, then the

Q -drift of U equals the P -drift of U + [U, l{L}(Z)] .

See also

• Stochastic exponential

Notes and References

1. Book: Jacod. Jean. Limit theorems for stochastic processes. Shiryaev. Albert Nikolaevich. 2003. Springer. 3-540-43932-3. 2nd. Berlin. 134–138. 50554399.
2. Larsson. Martin. Ruf. Johannes. 2019. Stochastic exponentials and logarithms on stochastic intervals — A survey. Journal of Mathematical Analysis and Applications. en. 476. 1. 2–12. 10.1016/j.jmaa.2018.11.040. 119148331 . free. 1702.03573.
3. Larsson. Martin. Ruf. Johannes. 2019. Stochastic exponentials and logarithms on stochastic intervals — A survey. Journal of Mathematical Analysis and Applications. en. 476. 1. 2–12. 10.1016/j.jmaa.2018.11.040. 119148331 . free. 1702.03573.